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Question #42553

Difficulty: 8/10 | 🟢 Free

The Question

An ideal monoatomic gas ( $n$ moles) is enclosed in a cylinder of cross-sectional area $A$ with perfectly insulating walls. The piston (area $A$ ) is highly conducting, has negligible heat capacity, and its outer surface acts as a perfect black body radiating heat into a surrounding vacuum at $0\text{ K}$ . The piston is pulled outward at a constant speed $v_0$ . If the initial volume and temperature of the gas are $V_0$ and $T_0$ respectively, which of the following expressions gives the temperature $T$ of the gas as a function of its volume $V$ ? (Let $\sigma$ be the Stefan-Boltzmann constant)

Options

A.
$\left[ T_0^{-3} \left( \frac{V}{V_0} \right)^2 + \frac{2 \sigma}{n R v_0} V \left( \frac{V}{V_0} - 1 \right) \right]^{-1/3}$
B.
$\left[ T_0^{-3} \left( \frac{V_0}{V} \right)^2 + \frac{2 \sigma}{3 n R v_0} V \left( 1 - \left(\frac{V_0}{V}\right)^3 \right) \right]^{-1/3}$
C.
$\left[ T_0^{-3} \left( \frac{V}{V_0} \right)^2 \right]^{-1/3}$
D.
$\left[ T_0^{-3} \left( \frac{V}{V_0} \right)^3 + \frac{3 \sigma}{2 n R v_0} V \left( \left(\frac{V}{V_0}\right)^2 - 1 \right) \right]^{-1/3}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Cyclic Process Geometry|Stefan-Boltzmann Power Law|Ideal Gas Assumptions

Why This Question Is Tricky

Common Mistakes

This question tests the ability to couple the First Law of Thermodynamics with a non-conservative, time-dependent radiative heat flux.

Students often fall into the trap of treating expansions as purely adiabatic ( $dQ=0$ ) or standard polytropic processes.

By introducing a constant-velocity piston, the problem forces the student to link the time differential $dt$ to the volume differential $dV$ , transforming the energy conservation equation into a non-linear Bernoulli differential equation.

The core misconception addressed is the assumption that thermodynamic paths are independent of the rate of the process; here, the piston's speed $v_0$ explicitly dictates the thermodynamic path.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Cyclic Process Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free